Mersenne #43 has been announced: 230,402,457 -
1. More details at MathWorld.com and Mersenne.org.
It has almost 10 million digits. One odd thing I only learned a few
months ago is that when p ± 1 isn't readily factorable (which I'll call
a nearby unfactorable prime), ten thousand digits
is considered amazing for a primality proof. Much of the progress in nearby unfactorable
primes has been made in the last few months, under listings Generalized
Lucas, Generalized Repunit,
Lehmer Number, Fibonacci
Number, Elliptic Curve,
and Lucas
Number. These numbers all have interesting algebraic qualities that lead
to a proof. Other prime numbers currently seem unprovably prime, these are the Probable
Primes.

Serhiy Grabarchuk greetings

Serhiy, who now runs Age of Puzzles,
sent in his annual holiday puzzle. This time, it's a field of Christmas Trees.

I've learned that wide diagrams don't work well on older browsers (why not
try Firefox?) and my new site layout.
Peter Esser found that the 196
one-sided heptominoes could fit into 14 lovely rectangles. I had to resize the original
GIF (with Irfanview), which
makes the lines a bit fuzzy. It would be nice to have this in a vector
format. Updated: Jeff Epler -- "I resized this in The
GIMP with a multi-step
process. My strategy for resizing was to first use "value propagate" to
make the
black lines two pixels thick. I also replaced the two tones of green with a single
cone. Then I rescaled the image to 50% nearest neighbor." [Oh, so that's how
to do it!]

Material added 21 Dec 05

Another Mersenne Prime

Under "Mersenne Exponent Test State" at PrimeNet,
there is a new unverified Mersenne Prime. It is probably getting tested right
now.

Do you like odd mechanisms? KMODDL is
a collection of mechanical models and related resources for teaching the principles
of kinematics--the geometry of pure motion. The core of KMODDL is the Reuleaux
Collection of Mechanisms and Machines, an important collection of 19th-century
machine elements held by Cornell's Sibley School of Mechanical and Aerospace
Engineering. For example, here is the Eccentric
Spur Gear.

The questions of the 2005 Putnam exam, which was held Dec 3, 2005, can be seen
at the Harvard math site. They
also have all the Putnam tests back to 1938, in case you'd like something to
solve during a holiday flight. For more Olympiad level mathematics, try the MathLinks
Forum.

Pentomino Oddities

In case I haven't mentioned it before, George Sicherman's Pentomino
Oddities page is well worth a look.

Here's a game played with 12 cards, numbered 0-11. The cards are distributed
randomly into two visible groups of cards, Group A and Group B, with the cards
being redealt if the Group A cards total less than 21. Players alternate turns.
Each turn involves swapping a card in Group A with a lower valued card in Group
B. Whoever makes the sum of the cards in Group A go under 21, loses. Pretty simple
game, right? Playing the game perfectly requires some deep group theory and Steiner
systems, described by David Joiner in his paper MINIMOGS
and Mathematical Blackjack.

4D Sudoku

Chris Lusby Taylor: The diagram below shows the SuDoku grid divided into its
3x3 blocks, which are then separated and skewed to show them as planes within
a 3x3x3x3 hypercube. This diagram should be familiar to anyone who has played
4D noughts and crosses.
From the hypercube, 3x3x3 cubes can be mentally assembled
by selecting a particular value for any one dimension. For instance, w=1 gives
a cube containing the three leftmost blocks. Any such cube can be sliced in three
ways to give a 3x3 block. For instance, w=1, z=3 is the top left block, w=1,
x=1 is the leftmost column. w=1, y=3 is also a valid slice, which I have colored
green. I have colored blue the set x=3, z=2, and red the set x=2, y=1. The puzzle
is to fill the 81 squares so that all such 3x3 slices contain every digit.
SuDoku
has previously been extended by coloring the squares. The Japanese call this
Colour Number Place. This adds sets of squares, such as the ones I color red,
which are all in the same position within a 3x3 block.
With all those extra constraints,
we can create a puzzle with fewer given numbers. Mine has just ten. It has a
unique solution which can be found by applying normal SuDoku logic, but over
six directions rather than three. It is not easy, but requires absolutely no
trial and error. (Answer and Solvers.)
[Some earlier 4D Sudoku by Guenter Stertenbrink are at magictour.]

Bob Hearn: I've attached
a couple of the latest kind of coin puzzle I've been generating. They both
have the added nice feature that there are a few tricky moves required near the
end. They take 48
moves (8 vertices), 66 moves (9 vertices),
and 118
moves (11 vertices) respectively,
and are definitely harder than they look. [[These are wonderful puzzles, well
worth printing out. His search also found a 173-move puzzle on 12 vertices,
and a 238-move puzzle on 13 vertices.]]

Hidden Points and Matchstick Graphs

Erich Friedman's latest Math
Magic contains several interesting problems,
including marksmen
duels, matchstick graphs, and hidden points. (Warning -- the second link
has the answer to this week's puzzle.) In the first figure below, discovered
by Erich Friedman, four points are hidden from every point. Also in the star,
discovered by Aron Fay, four points are hidden from each point. Erich found an
arrangement of 12 points where 3 points are hidden from each point. Can you find
this arrangement, or a different arrangement ... or find a construction that
hides 5 points from every point? (The first
image was drawn with TpX.)

Jonathan vos Post conjectures that every integer is a factor of a Tribonacci
number. For example, 10 is a factor of 35890, the 19th Tribonacci number,
by the method at MathWorld. The conjecture has been verified
to 500 -- seems
quite chaotic. Can anyone prove this conjecture, or verify it up to some big
number? Answer and Solvers.

Oskar van Deventer in Games Magazine, and River Crossing

The latest issue (Feb 06) of Games
Magazine has a great article about the many
puzzles of Oskar van Deventer. Oskar himself sent me sent me some nice photos
recently. Oskar: "Attached are three photos of large hands-on puzzles
that are on display in the Delft Science Museum. The person standing on the River
Crossing is Andrea Gilbert, the inventor
of this puzzle. The two other puzzles are Richard Tucker’s Hayling
Island Maze and my own Key
Maze, both made
by Wim Zwaan. According to Liesbeth van Hees, the creative director of the
science museum, the large puzzles are immensely popular with children." Oskar
is in the gray sweater below. Oskar's Planet's
puzzle is also recently available.

Magic Tours and more

Magic knight tours on the 12x12 board an many other topics are discussed in
the latest issue of the Games
and Puzzle Journal.

Floretions

Creighton Dement: I took part in the Congress/Bundestag youth exchange
program in 1992-93. My main area of work over the last two years has dealt with "floretions" (pronounced
floor-RET-ee-ons), named after mathematics Prof. K. Floret.

Cube Sudoku

Steve Schaefer: I've created a couple of new sudoku variants. The most notable
one is a cube sudoku that
uses all six sides of a cube with 4x4 faces. The "rows" and "columns" run
all the way around the cube, so that each type of block has 16 values. This seems
like an obvious configuration to me, so it's entirely possible that someone else
thought of it first, but I've never seen one that any one else made. That puzzle
was made by hand. I've now written a generator, but the first puzzle it spit
out is giving me fits as I try to solve it by hand. There are some new solve
rules on the surface of a cube and I may not have discovered them all. (Ariadne
picked up the slack in the solver that my generator uses.)

Highest Prime Gap Merit discovered

804212830686677669 is at the start of a prime
gap of length 1442. The merit of this gap is 34.98 -- the first improvement since 1999.

The 1/7th Ellipse -- updated

Jay Hall noticed that the six pairs of number in 1/7th, .142857... or (1, 4),
(4, 2), (2, 8), (8, 5), (5, 7) and (7, 1), all lie on an ellipse. Eric Weisstein
added this interesting object to MathWorld as
the 1/7th Ellipse.
Chris Lomont: "Out of curiousity, I found a lot more of these ellipses.
One with more points is the 1/7373 ellipse, 1/7373 = 0.00013653... which gives
seven points {0,0}, {0,1}, {1,3}, {3,0}, {3,5}, {5,6}, {6, 3} on an ellipse.
To get 8 points on a single ellipse I found the fraction 4111/3030303 works.
I'm yet to find more on a single ellipse. I'm unaware of any proof that it cannot
be done, although integer points on curves is very much studied."

Material added 21 Nov 05

Gordian's Knot puzzle now in stores

Last year, I wrote a column about modern
burr puzzles, and the phenomenal discovery by Frans de Vreugd of a puzzle
made of simple pieces that needs 28 moves to remove the first piece. I suggested
to Thinkfun/Binary Arts to talk to Frans about making it, and they did! Released
as Gordian's
Knot, it is now in stores. It's the ultimate fiddly puzzle. Bob Hearn:
"It is absolutely wonderful. I saw someone at MIT fiddling with one last week,
and had to go and get one right away. It's visually grabbing. And I can't stop
playing with it!" Another write-up is at Torturous
Burrs. Thinkfun has also
released a new puzzle by Ferdinand Lammertink,
Flip-Side.

Jordan Curve theorem proved

Jordan's Curve theorem proves that every closed, nonintersecting curve has
an inside and an outside. Jordan's original proof, though elegant, was not considered
a strict proof. An elaborate
computer system by teams in Japan and Poland has
developed, after 14 years of work, a strict proof which fills over 3000 pages.

Bridged DiTans and Triamonds

Bernd Karl Rennhak: I have some news about "Bridged
Polyforms".
That is a class of polyforms I investigated recently. The idea is not completely
new, but I pushed the thing a bit further. The bridging method is pretty generic
and can be applied to various polyforms. Basic forms are connected with vertex
and/or edges, where the vertex points need the bridges, which creates nice sets
of shapes. For the basic forms I used squares, equilateral triangles and isosceles
right triangles, still more to come. [[Update -- Oskar has made a very nice fractal
version of this.]]

Cool Illusion

A very
cool optical illusion involving an animation of pink dots is well worth
a look. jjartus notes that this illusion is accomplished by retinal fatigue --
for much the same reason, hospital gown green is the exact opposite color of
blood.

Knight Tour Sudoku

Dan Thomasson: Here are a couple Knight Tour Latin Square puzzles I designed
that should not be too difficult to figure out. After solving the squares, you
should be able to draw four separate 16-move closed knight tours in each square.
You can see more information about them on my website at www.borderschess.org/LatinKT-Problem.htm.

You can't have enough songs about Physics. To meet this requirements, there
is physicssongs.org.

No-guess 9-clue Geometric Sudoku

Bob Harris: Have finally found a 9x9 Latin Square Puzzle (a.k.a. jigsaw sudoku,
geometric sudoko, dusumoh) that requires no trial and error. It's at http://www.bumblebeagle.org/dusumoh/9x9/index.html.
I've only been searching with the "snakey" nonominoes, since previous
results suggest these have a higher density of such puzzles (no idea why, though).
Will start searching with other nonominoes at some point.

Toshi Kato: A NOBle puzzle should be clever, sophisticated, and worthy of puzzlemaster
Nob Yoshigahara. A Puzzle Design Competition started last week, and now the 13
entries are online (Japan
access). Only if you try to solve at least
one, then, you can vote best four puzzles with ranking. Please select best four
in thirteen puzzles for NOBle prize. Using a rating system, the five best solvers
will get a special wooden puzzle.

Amusingly, the 1980 issue of Games that started the
Games 100 was also the
first issue I was published in. Of note this year is the Puzzle of the Year,
Tipover, by James Stephens. Two years ago, it was River
Crossing, by Andrea
Gilbert.

Site Goals

Martin Gardner celebrates math
puzzles and
Mathematical Recreations. This site aims to do the same. If you've made
a good, new math puzzle, send it to ed@mathpuzzle.com.
My mail address is Ed Pegg Jr, 1607 Park Haven, Champaign, IL 61820.
You can join my moderated recreational mathematics email list at http://groups.yahoo.com/group/mathpuzzle/.

All material on this site is copyright
1998-2006 by Ed Pegg Jr. Copyrights
of submitted materials stays with the contributor and is used with
permission.